
By isolated Gorenstein singularity we mean a non Gorenstein ring \((R,\mathfrak {m})\)  which is Gorenstein over the punctured spectrum.

Let \((R,\mathfrak {m})\)  be a Cohen-Macaulay local ring of dimension \(d>1\) 
with isolated Gorenstein singularity and possessing a canonical module. In view of [1]}, we know that \(\operatorname{H}^{i}_{\mathfrak {m}}(\omega _R\otimes _R \omega _R^\ast )\ne 0\)  if and only if \(i\le 1\)  or \(i=d\) .

Let us reprove the claim \(\operatorname{H}^{i}_{\mathfrak {m}}(\omega _R\otimes _R \omega _R^\ast )=0\)  for all \(2\le i<\dim ( R)\) , by the methods developed in this paper. Indeed, \(\omega _R\)  is locally free on the punctured spectrum of \(R\) . Let \(i\ge 1\) . Then, in view of Lemma REF , we have that \(\operatorname{H}^{i}_{\mathfrak {m}}(\omega _R\otimes _R \omega _R^\ast )=\operatorname{Ext}^{i-1}_R(\omega _R,\omega _R)=0\)  for all \(i\)  where \(2\le i<\operatorname{depth}_R(\omega _R)=\dim (R)\) .

Any one-dimensional integral domain which is not Gorenstein is a Cohen-Macaulay ring with isolated Gorenstein singularity. For example,
let \(R:=k[[x^3,x^4,x^5]]\) .

Any two-dimensional normal domain which is not Gorenstein is a Cohen-Macaulay ring with isolated Gorenstein singularity. For example, \(R:=k[[x^3,x^2y,xy^2,y^3]]\) .

To see a 3-dimensional situation, let \(S := k[[x, y, z, u, v]]\)  and put \( R := S/(yv-zu, yu-xv, xz-y^2)\) , where \(k\)  is an algebraically closed field of characteristic
different from 2. By [2]}, \(R\)  is a 3-dimensional Cohen-Macaulay ring with isolated Gorenstein singularity.
In fact, its canonical module is isomorphic to
the ideal \((u, v)\) .


